Why Raindrops Fall Gently Instead of Hitting Like Bullets

Water pressure grows as a diver goes deeper, air rushing over a wing lifts an aircraft, and honey pours slowly while water splashes. Fluid mechanics explains all three with a few laws about pressure, flow and viscosity.

This lesson covers Pascal's law and pressure at depth, Bernoulli's theorem with lift and flow through constrictions, and Stokes' law, terminal velocity and Reynolds number.

**Pressure in a fluid at rest increases with depth as $p=p_0+\rho gh$, and Pascal's law states that a pressure applied to an enclosed fluid is transmitted undiminished to every part of it.

Pressure at depth:**

$$p=p_0+\rho gh$$

- $p_0$ is the surface pressure, usually atmospheric pressure, $1.01\times 10^5$ Pa
- Gauge pressure, $\rho gh$, depends only on depth, not on the shape of the container

Pascal's law. In a hydraulic lift, the same pressure acts on both pistons:

$$\frac{F_1}{A_1}=\frac{F_2}{A_2}\Rightarrow F_2=F_1\frac{A_2}{A_1}$$

Worked example 1 — a diver. At 10 m in sea water of density 1030 kg m${}^{-3}$, with $g=9.8$ m s${}^{-2}$:

$$p=1.01\times 10^5+1030\times 9.8\times 10=2.02\times 10^5\text{ Pa}$$

That is about twice atmospheric pressure.

Worked example 2 — a hydraulic lift. With pistons of 0.0020 m${}^2$ and 0.20 m${}^2$, a 150 N push on the small piston supports

$$F_2=150\times \frac{0.20}{0.0020}=15000\text{ N}$$

An everyday example. Hydraulic brakes in buses and cars use Pascal's law: a light press on the pedal becomes a large force on the brake pads at every wheel.

The substance. A hydraulic lift multiplies force, not energy — the large piston moves a much shorter distance, so the work done on both sides is the same.

**Bernoulli's theorem states that for steady, streamline flow of an ideal fluid, $p+\frac{1}{2}\rho v^2+\rho gh$ is constant along a streamline, so where a fluid speeds up, its pressure falls.**

$$A_1{v}_1=A_2{v}_2{p}_1+\frac{1}{2}\rho v_1^2+\rho g{h}_1=p_2+\frac{1}{2}\rho v_2^2+\rho g{h}_2$$

- The equation of continuity says a fluid speeds up where a pipe narrows
- Bernoulli's theorem is conservation of energy per unit volume of fluid

Flow through a constriction. Water flows at 2.0 m s${}^{-1}$ through a pipe of area 0.010 m${}^2$ into a narrow section of area 0.0025 m${}^2$:

$$v_2=\frac{0.010\times 2.0}{0.0025}=8.0{\text{ m s}}^{-1}{p}_1-p_2=\frac{1}{2}\times 1000\times (8.0^2-2.0^2)=3.0\times 10^4\text{ Pa}$$

A Venturi meter uses exactly this pressure drop to measure flow speed.

Lift on an aerofoil. A wing's shape and tilt make air flow faster over its upper surface than below, so the pressure above is lower. If air of density 1.2 kg m${}^{-3}$ moves at 70 m s${}^{-1}$ above a wing and 60 m s${}^{-1}$ below it:

$$\Delta p=\frac{1}{2}\times 1.2\times (70^2-60^2)=780\text{ Pa}$$

so every square metre of wing feels an upward force of 780 N.

An everyday example. A paint spray gun or perfume atomiser draws liquid up a tube because fast-moving air across the top of the tube lowers the pressure there.

The substance. Bernoulli's theorem assumes a non-viscous, incompressible fluid in steady flow — real fluids lose some energy to viscosity, so the actual pressure drop is slightly larger.

**Stokes' law gives the viscous drag on a small sphere as $F=6\pi \eta rv$; when drag and buoyancy balance the weight, the sphere falls at a steady terminal velocity, $v_t=\frac{2r^2(\rho -\sigma )g}{9\eta }$, while Reynolds number, $R_e=\frac{\rho vd}{\eta }$, tells whether flow is streamline or turbulent.

Viscosity** is a fluid's internal friction, measured by the coefficient $\eta$ in Pa s.

Terminal velocity. For a sphere of density $\rho$ in a fluid of density $\sigma$, weight equals buoyant force plus drag:

$$\frac{4}{3}\pi r^3\rho g=\frac{4}{3}\pi r^3\sigma g+6\pi \eta r{v}_t\Rightarrow v_t=\frac{2r^2(\rho -\sigma )g}{9\eta }$$

Worked example 1 — a raindrop. A drop of radius 0.10 mm falls through air with $\eta =1.8\times 10^{-5}$ Pa s, neglecting the density of air:

$$v_t=\frac{2\times (1.0\times 10^{-4}{)}^2\times 1000\times 9.8}{9\times 1.8\times 10^{-5}}=1.2{\text{ m s}}^{-1}$$

Reynolds number. $R_e=\frac{\rho vd}{\eta }$ is low, below about 1000, for streamline flow and high for turbulent flow; high speed, wide pipes and low viscosity favour turbulence.

Worked example 2. Water, with $\eta =1.0\times 10^{-3}$ Pa s, flows at 0.050 m s${}^{-1}$ through a pipe 1.0 cm wide:

$$R_e=\frac{1000\times 0.050\times 0.010}{1.0\times 10^{-3}}=500$$

so the flow is streamline.

An everyday example. Smoke rising from an incense stick starts as a smooth streamline and then breaks into turbulent swirls as it speeds up.

The substance. Terminal velocity grows with the square of the radius — a drop twice as wide falls four times as fast, which is why fine drizzle drifts while big monsoon drops fall hard.